Shrinking targets problems for flows on homogeneous spaces

Abstract

We study shrinking targets problems for discrete time flows on a homogenous space $\Gamma \backslash G$ with $G$ a semisimple group and $\Gamma$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.

31 pages; Corollary 1.2 is now proved in full generality; removed the assumption that $\{\mu (B_m)\}$ is non-increasing in Proposition 5.4; typos corrected

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